We show that every n × n matrix is generically a product of n/2 + 1 Toeplitz matrices and always a product of at most 2n + 5 Toeplitz matrices. The same result holds true if the word ‘Toeplitz’ is replaced by ‘Hankel,’ and the generic bound n/2 + 1 is sharp. We will see that these decompositions into Toeplitz or Hankel factors are unusual: We may not, in general, replace the subspace of Toeplit...

Abstract. We investigate the simplest class of hyperdeterminants defined by Cayley in the case of Hankel hypermatrices (tensors of the form Ai1i2...ik = f(i1+i2+· · ·+ik)). It is found that many classical properties of Hankel determinants can be generalized, and a connection with Selberg type integrals is established. In particular, Selberg’s original formula amounts to the evaluation of all Ha...

To a sequence (sn)n≥0 of real numbers we associate the sequence of Hankel matrices Hn = (si+j), 0 ≤ i, j ≤ n. We prove that if the corresponding sequence of Hankel determinants Dn = detHn satisfy Dn > 0 for n < n0 while Dn = 0 for n ≥ n0, then all Hankel matrices are positive semi-definite, and in particular (sn) is the sequence of moments of a discrete measure concentrated in n0 points on the ...

This paper presents a fast algorithm for bidiagonalizing a Hankel matrix. An m×n Hankel matrix is reduced to a real bidiagonal matrix in O((m+ n)n log(m+ n)) floating-point operations (flops) using the Lanczos method with modified partial orthogonalization and reset schemes to improve its stability. Performance improvement is achieved by exploiting the Hankel structure, as fast Hankel matrix–ve...

An extensive literature exists describing various techniques for the evaluation of Hankel determinants. The prevailing methods such as Dodgson condensation, continued fraction expansion, LU decomposition, all produce product formulas when they are applicable. We mention the classic case of the Hankel determinants with binomial entries 3k+2 k and those with entries 3k k ; both of these classes o...

We present a solution to scale spectral algorithms for learning sequence functions. We are interested in the case where these functions are sparse (that is, for most sequences they return 0). Spectral algorithms reduce the learning problem to the task of computing an SVD decomposition over a special type of matrix called the Hankel matrix. This matrix is designed to capture the relevant statist...

We present a solution to scale spectral algorithms for learning sequence functions. We are interested in the case where these functions are sparse (that is, for most sequences they return 0). Spectral algorithms reduce the learning problem to the task of computing an SVD decomposition over a special type of matrix called the Hankel matrix. This matrix is designed to capture the relevant statist...

This paper presents an O(n2 log n) algorithm for computing the symmetric singular value decomposition of square Hankel matrices of order n, in contrast with existing O(n3) SVD algorithms. The algorithm consists of two stages: first, a complex square Hankel matrix is reduced to a complex symmetric tridiagonal matrix using the block Lanczos method in O(n2 log n) flops; second, the singular values...

We apply and extend some well known and some recent tech niques from algebraic residue theory in order to relate to each other two major subjects of algebraic and numerical computing that is computa tions with structured matrices and solving a system of polynomial equa tions In the rst part of our paper we extend the Toeplitz and Hankel structures of matrices and some of their known properties ...

This paper investigates the eigenstructure of Hankel operators for nonlinear systems. It is proved that the variational system and Hamiltonian extension can be interpreted as the Gâteaux differentiation of dynamical input-output systems and their adjoints respectively. We utilize this differentiation in order to clarify the eigenstructure of the Hankel operator, which is closely related to the ...